Finite Group Databasebeta

← #19464

Subgroups — #19464

Order 256.

329 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3230C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic
3C2×#461C211yesmaximal, characteristic
4#2974C211yesmaximal, characteristic
5#1699C212yesmaximal, complemented
6#1693C212yesmaximal, complemented
7#1679C212yesmaximal, complemented
8#1673C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic, complemented
10#1556C211yesmaximal, characteristic, complemented
11#1552C211yesmaximal, characteristic, complemented
12C22×SD16—21no—
13#533C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#500C2212yescomplemented
17#487C2212yescomplemented
18#484—21no—
19#484—21no—
20#484—21no—
21#477—22no—
22#474C2211yescharacteristic
23#474C2211yescharacteristic
24#461—21no—
25#461—21no—
26#461C2211yescharacteristic
27#461C2211yescharacteristic
28#461C2212yes—
29#461C2212yes—
30#450—21no—
31#440—21no—
32#440—21no—
33#436C2212yes—
34#435—22no—
35C8×D4C2212yes—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#431C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×#103—21no—
41#407—21no—
42C2×C22⋊C8—21no—
43#335C2212yescomplemented
44#334C2212yescomplemented
45#332C2212yescomplemented
46#332C2212yescomplemented
47#332C2212yescomplemented
48#332C2212yescomplemented
49C22×Q8—21no—
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C2×SD16—22no—
53C2×SD16—22no—
54C2×SD16—21no—
55C2×SD16—21no—
56C2×SD16—44no—
57C2×M16—22no—
58C22×C8—21no—
59C22×C8—22no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8C2312yes—
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66#125—21no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4D411yescharacteristic
74C4×D4D411yescharacteristic
75C4×D4C2311yescharacteristic
76C4×D4C2311yescharacteristic
77#117—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C42C2311yescharacteristic
83#107—22no—
84#107—22no—
85#107—21no—
86#107—21no—
87C4⋊C8—21no—
88C4⋊C8D412yes—
89C4⋊C8C2312yes—
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2312yes—
92C4⋊C8C2311yescharacteristic
93#103—22no—
94#103—22no—
95#103—22no—
96#103—22no—
97#103—21no—
98#103—21no—
99#102—21no—
100#102—21no—
101C22⋊C8—21no—
102C22⋊C8—21no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105C22⋊C8D411yescharacteristic
106C22⋊C8D411yescharacteristic
107#97—21no—
108#97—21no—
109C4×C8C2312yes—
110C24—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—41no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—41no—
120C2×D4C2×D411yescharacteristic
121C2×D4C2×D411yescharacteristic
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—82no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4#4111yescharacteristic
132SD16—42no—
133SD16—44no—
134SD16—42no—
135SD16—44no—
136M16—82no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—42no—
142C2×C8—82no—
143C2×C8C2×D412yes—
144C2×C8C2×D412yes—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4C2×D411yescharacteristic
157C4⋊C4C2×D411yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—41no—
163C42—41no—
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42#4111yescharacteristic
167C42C2411yescharacteristic
168C23—21no—
169C23—21no—
170C23—42no—
171C23—42no—
172C23—81no—
173C23#12111yescharacteristic
174Q8—42no—
175Q8—41no—
176Q8—41no—
177Q8—42no—
178Q8—42no—
179Q8—82no—
180D4—21no—
181D4—21no—
182D4—21no—
183D4—42no—
184D4AGL(1,ℤ/8)12yes—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4#12111yescharacteristic
201C2×C4#12111yescharacteristic
202C2×C4#12111yescharacteristic
203C2×C4C22×D411yescharacteristic
204C2×C4C22×D411yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C8—22no—
207C8—22no—
208C8—42no—
209C8—41no—
210C8—41no—
211C8—42no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—84no—
221C22#47312yes—
222C22C2×#12111yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—81no—
230C4—82no—
231C4—81no—
232C4C2×AGL(1,ℤ/8)11yescharacteristic
233C4#57511yescharacteristic
234C4#57712yes—
235C2—22no—
236C2—42no—
237C2—81no—
238C2#304211yescharacteristic
239C2#305511yescharacteristic
240C2C2×#47311yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3230C2semidirect
5. #1699C2semidirect
6. #1693C2semidirect
7. #1679C2semidirect
8. #1673C2semidirect
9. #1573C2semidirect
10. #1556C2semidirect
11. #1552C2semidirect
16. #500C22semidirect
17. #487C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #332C22semidirect
46. #332C22semidirect
47. #332C22semidirect
48. #332C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#32302
#31102
C2×#4612
#29742
#16992
#16932
#16792
#16732
#15732
#15562
#15522